Exploration of the effects of Coriolis force and thermal radiation on water-based hybrid nanofluid flow over an exponentially stretching plate

Hybrid nanofluids’ enhanced thermophysical properties make them applicable in a plethora of mechanical and engineering applications requiring augmented heat transfer. The present study focuses on a three-dimensional Copper-Aluminium Oxide \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( Cu\text{- }Al_{2}O_{3}\right)$$\end{document}Cu-Al2O3-water based hybrid nanofluid flow within the boundary layer with heat transfer over a rotating exponentially stretching plate, subjected to an inclined magnetic field. The sheet rotates at an angular velocity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document}Ω and the angle of inclination of the magnetic field is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document}γ. Employing a set of appropriate similarity transformation reduces the governing PDEs to ODEs. The resulting ODEs are solved with the finite difference code with Shooting Technique. Primary velocity increases at large rotation but the secondary velocity reduces as the rotation increases. In addition, the magnetic field is found to oppose the flow and thereby causing a reduction in both the primary and secondary velocities. Increasing the volume fraction reduces the skin friction coefficient and enhances the heat transfer rate.

www.nature.com/scientificreports/ has no significant influence on all transport phenomena. When the liquid motion speed is small in comparison to the rotation speed, the Coriolis effect becomes negligible, which is why the Coriolis effect is not readily experienced on earth. From past decades, numerous researchers are exploring the Coriolis force impact on diverse liquid streams [32][33][34][35][36][37][38] and in each of the studies, Coriolis effect was found to be significant on the flow velocity.
It is important to note that based on the available research literature, no research has been done on the simultaneous impact of the heat radiation and Coriolis force on the water transporting copper and alumina nanoparticles on a rotating exponentially stretching plate. Hence, this study is novel and has practical significance in mathematics and engineering, and will open a space for further research. The following research questions are answered in this study; 1. How does increasing Coriolis effect impact the flow of copper-alumina-water-based hybrid nanofluid subjected to thermal radiation? 2. How does raising the the size of the MF strength affect the skin friction and heat transfer rate coefficients in the flow of copper-alumina-water-based hybrid nanofluid flow 3. How does increasing inclination angle affect the flow of water-based hybrid nanofluid? 4. How does increasing volume fraction affect heat transfer rate in the flow of water-based hybrid nanofluid?

Governing equations and methodology
This study analyses a 3D boundary layer flow of electrically conducting water-based hybrid nanofluid past an exponentially stretched sheet. Figure 1 shows the set-up of flow configuration. The sheet rotates at an angular velocity and the flow is steady, laminar and incompressible. An inclined MF of strength B is applied to the surface at an angle γ . Following the formulations of Nayak et al. 39 and Oke et al. 35 , the equations governing the flow is given in Eqs. (1)(2)(3)(4); The boundary and initial conditions are given in Eqs. (5) and (6); u = U w = U 0 e x+y , v = V w = U 0 e x+y , w = 0, T = T w = T ∞ + T 0 e 2(x+y) at z = 0,  www.nature.com/scientificreports/ The effective dynamic viscosity µ hnf and effective density ρ hnf of the hybrid nanofluid 21 are defined in Eq. (7) below; where φ is the overall volume fraction defined as φ = φ 1 + φ 2 10 . The effective thermal diffusivity α hnf and the specific heat capacity ρc p hnf [40][41][42] are defined as shown in Eqs. (8) and (9); The skin friction and heat transfer rate along the x-and z-axes are the quantities of engineering relevance, and they are given in Eq. (10) as respectively. The shear stress τ along the x-and y-directions on the wall and the wall heat flux q w are defined as the following quantities evaluated at the wall ( i.e. z = 0); Methodology. The partial differential equations (1-4) with the initial and boundary conditions (5 and 6) are nondimensionalised using the similarity variables given in Eqs. (11)(12)(13) . and s 1 , s 2 , s 3 are chosen to satisfy the boundary conditions at η ∞ ; This coupled system of ordinary differential equations is highly non-linear and cannot be solved analytically. The semi-analytical method of solutions can be used but they require a lot of computation time 43 . Hence, the numerical methods provide a more efficient and computationally-economical approach to finding the solutions. The solution of Eqs.  Table 1.

Discussion of results
The outcomes associated with the heat transfer in water-based hybrid nanofluid flow along a rotating and exponentially-stretching plate are hereby discussed. The governing equations are modelled with the presence of Coriolis force and MF. The study elucidates the significance of involved controlling numerous somatic factors in the modelling equations using graphs and tables. Variation in drag coefficient f ′′ (0) and g ′′ (0) and the Nusselt number −� ′ (0) for varying pertinent parameters are tabulated in Table 2.
Practically, increase in M and K are consequences of increased magnetic field strength and surface rotation respectively. The presence of magnetic field around the electrically-conducting fluid tends to oppose flow while rotation propels the flow forward in the direction of the flow. Raising the values of M and K improves f ′ (0) but a conflicting trend is seen for growing values of volume fraction, R and γ . Further, an upsurge in nanoparticle volume fraction φ increases the rate of heat transfer −� ′ (0) but a conflict trend is seen for growing values of M, K, R and γ . By raising the volume fraction of the nanoparticles, the thermal conductivity of the nanofluid is improved and therefore the thickness of the thermal boundary layer grows, resulting in an increasing rate of (21) . www.nature.com/scientificreports/ heat transfer. The rate of thermal heat transfer is substantially influenced by thermal radiation. When the volume of nanoparticles grows, the heat transfer rate falls as thermal radiation rises. Figures 2 and 3 are designed to explore the role of rotation parameter K on primary and secondary velocity profiles. Here, to obtain the variation of the pertinent profiles the parameters are kept fixed as M = 2, φ 1 = φ 2 = 0.01, Gr = 2, Pr = 6.9, R = 2, and γ = π/6, while the values of rotation parameter K = 0.001, 0.5, 1, 1.5, 2 is varied. The Coriolis force becomes stronger by increasing the values of K, which leads www.nature.com/scientificreports/ to an upsurge in the primary velocity profile. Further, increasing rotation parameter K causes a decline in the secondary velocity profile. This is all because of the significant influence of Coriolis force along with the stretching influence. The inertia force accountable for the deviation of the trajectory of liquid flow along a spinning surface is known as the Coriolis force and it becomes stronger by raising K, which leads to upsurge in primary velocity profile. Further, larger value of rotation parameter K reduces the secondary velocity profile. Physically, when K becomes larger, the rotation effects take precedence over the stretching effects, slowing the flow velocity. This is all because of the significant influence of Coriolis force along with the stretching influence. Figures 4 and 5 elucidate the leverage of M on both primary and secondary velocity profiles. All parameters are kept fixed as φ 1 = φ 2 = 0.01, K = 0.1, Gr = 2, Pr = 6.9, R = 2, and γ = π/6 while magnetic parameter M = 0.001, 0.5, 1, 1.5, 2 is varied to examine its consequence on the flow fields. Increase in M inhibits the flow and thereby causes a reduction in the velocity profiles. The presence of an MF in the flow region has been shown to slow down flow velocity. The magnetic force adds a layer of resistance to the flow and slows down the flow. The existence of a transverse MF induces the Lorentz force, which acts as a retarding force on the velocity field of base liquid and nanoparticles. As a result, as seen in the figures, this negative body force slows the boundary layer flow and inhibits momentum diffusion.
Variation of primary and secondary velocities and thermal profiles for various values of φ 1 and φ 2 are shown in Figs. 6, 7 and 8 . An increase in values of φ 1 and φ 2 boosts the primary and secondary velocities but declines the  www.nature.com/scientificreports/ thermal profiles. Here, the parameters are kept fixed as M = 2, K = 0.1, Gr = 2, Pr = 6.9, R = 2, and γ = π/6 while the values of nanoparticle volume fraction φ 1 = φ 2 is varied between 0.01 to 0.005. Increasing solid volume fractions enhances the thickness of the boundary layer. As a result, the fluid will flow faster which increases primary, and secondary velocity profile. Addition of solid nanoparticles to the base fluid will gradually decline the thermal distribution due to decrease in the thickness of the related boundary layer. Figures 9 is depicted to elucidate the influence of R on thermal profile. Here, the parameters are kept fixed as M = 2, K = 0.1, Gr = 2, Pr = 6.9, φ 1 = φ 2 = 0.01, and γ = π/6 while, the values of R is varied between 1 to 6. The effect of thermal radiation increases the temperature profile, as seen in this figure. Radiative heat transmission is less effective than conductive heat transport, lowering the buoyancy force. High R delivers more heat to functional nanofluids, which indicates an increment in thermal profile. The variation is more gradual than when the radiation parameter is at a lower value. When the radiation parameter is set to a higher value, the fluid is heated more and more, increasing the thermal profile. Figures 10 and 11    www.nature.com/scientificreports/ increased viscous force. When the angular velocity is increased, the average kinetic energy is predicted to grow as well. This gradually causes the fluid velocity to decline.

Conclusion
The heat transference in water-based hybrid nanofluid flow over a rotating exponentially stretching plate is explored in this analysis. The governing equations are modelled in the occurrence of Coriolis force and MF. A reformulation of the governing equations in their dimensionless forms using similarity transformation is first carried out and the resulting equations are solved using the finite difference scheme. The study elucidates the significances of involved controlling numerous somatic factors in the modelling equations with the graphs and tables. The most important outcomes of this study are: • The rise in rotation parameter results in stronger Coriolis force, which leads to upsurge in primary velocity profile but declines the secondary velocity profile.  www.nature.com/scientificreports/ • Increase in MF parameter declines the flow in both velocity profiles due to the presence of a transverse MF that generates the Lorentz force, which acts as a inhibiting force on the velocity field. • The rise in the MF inclination angle in the region improves molecular movements and interactions, resulting in increased viscous force as a result primary and secondary flow velocity decreases. • The escalation in values of radiation parameter delivers more heat to functional nanofluids which augments the heat transfer. • Increasing MF strength and rotation parameters improves the skin friction coefficient conflict trend is seen for growing values of volume fraction, radiation parameter and MF inclination angle. • The upsurge in values of volume fraction improves the heat transfer rate but the opposite is seen for growing values of MF inclination angle, MF strength, rotation and radiation parameters.